# Term Exam 1

University of Toronto, November 16, 2004

This document in PDF: Exam.pdf

Solve 5 of the following 6 problems. Each problem is worth 20 points. If you solve more than 5 problems indicate very clearly which ones you want graded; otherwise a random one will be left out at grading and it may be your best one! You have an hour and 50 minutes. No outside material other than stationary is allowed.

Problem 1. Let be an arbitrary topological space. Show that the diagonal , taken with the topology induced from , is homeomorphic to . (18 points for any correct solution. 20 points for a correct solution that does not mention the words inverse image'', open set'', closed set'' and/or neighborhood''.)

Problem 2. Let be a connected metric space and let and be two different points of .

1. Prove that if then the sphere of radius around , , is non-empty.
2. Prove that the cardinality of is at least as big as the continuum: .

Problem 3.

1. Define  is completely regular''.
2. Prove that a topological space can be embedded in a cube (a space of the form , for some ) iff it is completely regular.

Problem 4.

1. Define  is (normal)''.
2. For the purpose of this problem, we say that a topological space is if whenever and are disjoint closed subsets of , there exist open sets and in so that , and . Prove that if is then it is also .

Problem 5. The diameter'' of a metric space is defined to be .

1. Prove that a compact metric space has a finite diameter.
2. Prove that if is a compact metric space, then there's a pair of points so that .

Problem 6. If is a sequence of continuous functions such that for each , show that is continuous at uncountably many points of .

Good Luck!

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Dror Bar-Natan 2004-11-16